A fast marching level set method for monotonically advancing fronts.

Abstract

A fast marching level set method is presented for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial differential equation for a propagating level set function and use techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. This paper describes a particular case of such methods for interfaces whose speed depends only on local position. The technique works by coupling work on entropy conditions for interface motion, the theory of viscosity solutions for Hamilton-Jacobi equations, and fast adaptive narrow band level set methods. The technique is applicable to a variety of problems, including shape-from-shading problems, lithographic development calculations in microchip manufacturing, and arrival time problems in control theory. This paper describes and tests a numerical algorithm for tracking the evolution of interfaces. The technique applies in the case of a front propagating normal to itself with a speed F that depends only on position and is always either positive or negative. The applications of such a technique include some global illumination problems and problems from control theory, as well as surface advancement in lithographic development and isotropic etching and deposition in the manufacturing of microelectronic structures. This scheme was first described in ref. 1; this paper presents the details of this scheme and shows results and timings.Background

Regularization-based identification for level set equations

“…Let Γ D (t), t ∈ [0, T ] be the position data of the surface and let Ω D− (t) and Ω D+ (t) be the inner and outer volumes of the surface Γ D (t), respectively. Then, we can numerically construct a level set function φ D ( · , t) whose zero level set and zero sublevel set coincide with Γ D (t) and Ω D− (t), respectively [24,27]. Our goal is to find the speed function F ∈ F such that the zero level set of the solution of (3) matches Γ D as well as possible.…”

Level Set Method for Simulating the Interface Kinematics: Application of a Discontinuous Galerkin Method

“…It means that the re-initialization procedure is supposed to affect the level set function except its zero iso-value. The level set function can be re-initialized either by performing a geometrical technique such as the fast marching method introduced in [23], or operating the re-initialization over the field of level set function. The latter approach which was introduced in [24], is followed in the present research.…”

Cortical surface segmentation and mapping

“…The desired function T(x) is the time it takes to travel from Γ to x; because unit speed propagation is assumed, T is equivalent to geodesic distance. This equation can be solved using the fast marching (FM) method originally developed in Sethian (1996) and extended to triangulated domains in Kimmel and Sethian (1998).…”